Convex Optimization 2026, 04 - Generalized Inequalities
A cone is proper, if it's convex, closed, solid (meaning its interior is nonempty) and pointed (meaning it contains no line) (2.4.1, p 1). It defines a generalized inequality, partially ordering on ℝn inherits properties from standard ordering on ℝ. Given a proper cone K, x ≼K y ⇔ y - x ∈ K, with strict partial ordering x ≺K y ⇔ y - x ∈ int K. A generalized inequality ≼K is
- preserved under addition
- transitive
- preserved under nonnegative scaling
- reflexive
- antisymmetric
- preserved under limits
≺K satisfies, among other things
- x ≺K y ⇒ x ≼K y
- x ≺K y ⇒ x + u ≼K y + v ∀ u ≼K v
- x ≺K y ⇒ αx ≺K αy ∀ α > 0
- x ⊀K x
- x ≺K y ⇒ ∃ u, v < ε: x + u ≺K y + v
(2.4.1). This implies the existences of minimum and minimal elements. A minimum element of a set S is one that for all y ∈ S: x ≼K y. The maximum element is defined in parallel. A minimal element is defined as x ∈ S: y ∈ S, y ≼K x ⇔ y = x. Similarly, the maximal element (2.4.2, p. 2). By set notation, a minimum element of S is S ⊂ x + K.